Simple enough ?
Probability
Level
3
What is the sum of all the 5-digit natural numbers which can be created by using the digits 1,3,5,7 and 9 with no digit repeating?
The answer is 6666600.
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1 solution
Let us fix any one number in the digits place of the 5-digit number. In this case, there are 4 ! ways to arrange the remaining 4 digits. So, there will be 4 ! of each 1 , 3 , 5 , 7 , 9 in the digits place in the given sum. Similarly, there will be 4 ! ways to arrange it in the tens place, hundred place, thousands place and ten-thousands place.
Let S n denote the sum of all numbers possible. S n = ( 4 ! × 9 + 4 ! × 7 + . . . + 4 ! × 1 ) + ( 4 ! × 9 0 + . . . . . 4 ! × 1 0 0 0 0 ) S n = 4 ! ( 9 × ( 1 + 1 0 + 1 0 0 + 1 0 0 0 + 1 0 0 0 0 ) + . . . 1 × ( 1 + 1 0 + 1 0 0 + 1 0 0 0 + 1 0 0 0 0 ) S n = 4 ! ( 9 + 7 + 5 + 3 + 1 ) ( 1 1 1 1 1 ) S n = 2 4 × 2 5 × 1 1 1 1 1 S n = 6 6 6 6 6 0 0